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**MODELING REAL LIFE**

The table shows the speeds of a car as it travels through an intersection with a stop sign. What type of function can you use to model the data? Estimate the speed of the car when it is 20 yards past the intersection.

\[
\begin{array}{|c|c|}
\hline
\text{Displacement from sign (yards), } x & \text{Speed (miles per hour), } y \\
\hline
-100 & 40 \\
\hline
-50 & 20 \\
\hline
-10 & 4 \\
\hline
0 & 0 \\
\hline
10 & 4 \\
\hline
50 & 20 \\
\hline
100 & 40 \\
\hline
\end{array}
\]

An absolute value function can be used to model the data. The estimated speed of the car 20 yards from the stop sign is _____.

Answer :

To model the data given by the speeds of a car at different displacements from a stop sign, we use the following observations and approach:

1. Identify the Pattern: By observing the data provided in the table:

| Displacement from sign (yards), x | Speed (miles per hour), y |
|------------------------------------|----------------------------|
| -100 | 40 |
| -50 | 20 |
| -10 | 4 |
| 0 | 0 |
| 10 | 4 |
| 50 | 20 |
| 100 | 40 |

We notice that the speed values are symmetric around the origin (0, 0). This suggests that an absolute value function may be appropriate to model the data.

2. Formulating the Absolute Value Function: The general form of an absolute value function is:

[tex]\[
y = a \cdot |x - h| + k
\][/tex]

Given the symmetry around [tex]\( x = 0 \)[/tex], it is reasonable to assume [tex]\( h = 0 \)[/tex]. So our function simplifies to:

[tex]\[
y = a \cdot |x| + k
\][/tex]

3. Determining the Coefficients (a and k): We can use the given data points to determine the coefficients.

- At [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]. This implies [tex]\( k = 0 \)[/tex].
- Now, to find the value of [tex]\( a \)[/tex], we can use another point. Let's use the point [tex]\( (10, 4) \)[/tex]:

[tex]\[
4 = a \cdot |10| + 0 \implies 4 = 10a \implies a = \frac{4}{10} = 0.4
\][/tex]

Therefore, our function is:

[tex]\[
y = 0.4 \cdot |x|
\][/tex]

4. Estimating the Speed at 20 Yards: To estimate the speed of the car 20 yards from the stop sign, we evaluate the function for [tex]\( x = 20 \)[/tex]:

[tex]\[
y = 0.4 \cdot |20| = 0.4 \times 20 = 8
\][/tex]

Thus, the estimated speed of the car when it is 20 yards past the intersection is 8 miles per hour.

Therefore, a(n) absolute value function can be used to model the data. The estimated speed of the car 20 yards from the stop sign is 8 miles per hour.

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Rewritten by : Barada